Player A begins by placing a checker in the lower left-hand corner
of a checkerboard (8 by 8 squares). Player B places a checker one
square to the right or one square up or one square diagonally up and
to the right of Player A's checker... Would you rather be Player A or
Player B?

Place n distinct points on the circumference of a circle and draw all
possible chords through pairs of these points. Assume that no three of
these chords pass through the same point. Find and solve the recurrence
relation for the number of interior intersection points formed inside the
circle.

Use mathematical induction to prove that for any positive integer n,
if any one square is removed from a 2^n x 2^n checkerboard, then the
remaining squares can be completely (and exactly) covered with
L-shaped pieces composed of three squares.

In a graph with infinite "points," if we colour the lines with two colors
we'll have either a red or a blue infinite chain of lines, an infinite
number of points, all of them joined to each other with the same
colour...

I have heard that Graham's number is the largest number with
mathematical use. I have seen it expressed in arrow notation but
that does not give me a sense of how large it is. Is there a way to
express the number of digits it contains?

A company owner writes in for help cost-efficiently assigning tasks to different employees when each one commands her own fee for every job. Invoking a little graph theory, Doctor Jacques introduces the Hungarian algorithm and walks through an application to an example assignment.

Every road in Sikinia is one way. Every pair of cities is connected
by exactly one direct road. Show that there exists a city which can
be reached from every other city either directly or via at most one
other city.

Given the equation 5y - 3x = 1, how can I find solution points where x
and y are both integers? Also, how can I show that there will always be
integer points (x,y) in ax + by = c if a, b and c are all integers?

I have 14 invoice values and 2 report totals. The sum of the invoices for
each report should be the report total. Is there a fast way of finding
which invoices go with each report, or determining if it is even
possible?

A very challenging math game provides the background for a discussion
of how to find the winning strategy in 'reduced state' games, where
players attempt to force a final outcome after a series of moves.